BRATISLAVA MATHEMATICS CORRESPONDENCE SEMINAR
Faculty of Mathematics and Physics of Comenius University
Union of Slovak Mathematicians and Physicists
Centre of Spare Time - IUVENTA

BMCS - Problem set of the 2nd spring series 1997/98

Circles

  1. Two circles k1 and k2 intersect in points A, B. A point C lies on the line AB. Let KL, MN are chords of circles k1, k2 respectively, such that the point C lies on both KL and MN. Prove that all points K,L,M,N lie on some circle.

  2. ABCD is a convex quadrangle. Prove that if the diagonal AC has the same points of tangency to the circles inscribed in triangles ABC and ADC, then also the diagonal BD has the same points of tangency to the circles inscribed in triangles ABD and BCD.

  3. 1998 circles from all over the world met at the curling championship. Each pair of circles played exactly one game. Each game was finished by a victory of one of the playing circles. Prove that we can line up the circles in such order, that for each pair of subsequent circles, the front circle won a game with the rear one.

  4. The circles k1 and k2 do not intersect. A line p is an "outside" tangent to both circles. A line q is an "inside" tangent to both circles. A and B are points of tangency to k1 (for lines p and q respectively) and points C and D are points of tangency to k2 (for lines p and q respectively). Prove that the intersection of lines AB and CD lies on the line connecting the centers of circles k1 and k2.

  5. Prove that the intersections of the opposite sides of a hexagon, which is inscribed in a circle (in case they are not parallel) lie on one line.